On Testing Whether an Embedded Bayesian Network Represents a Probability Model

TL;DR

This paper investigates the complexity of testing whether embedded Bayesian networks with hidden variables accurately represent probability models, revealing exponential difficulty in general but polynomial solutions for tree-structured networks.

Contribution

It demonstrates the computational complexity of model validation and provides an efficient algorithm for tree-structured Bayesian networks with hidden variables.

Findings

Testing general models requires exponential independence evaluations.

Polynomial-time testing is possible for networks with a tree skeleton.

An algorithm is provided to construct or recognize such tree-structured models.

Abstract

Testing the validity of probabilistic models containing unmeasured (hidden) variables is shown to be a hard task. We show that the task of testing whether models are structurally incompatible with the data at hand, requires an exponential number of independence evaluations, each of the form: "X is conditionally independent of Y, given Z." In contrast, a linear number of such evaluations is required to test a standard Bayesian network (one per vertex). On the positive side, we show that if a network with hidden variables G has a tree skeleton, checking whether G represents a given probability model P requires the polynomial number of such independence evaluations. Moreover, we provide an algorithm that efficiently constructs a tree-structured Bayesian network (with hidden variables) that represents P if such a network exists, and further recognizes when such a network does not exist.